Hume-Rothery rules, named after William Hume-Rothery, are a set of basic rules that describe the conditions under which an element could dissolve in a metal, forming a solid solution. There are two sets of rules; one refers to substitutional solid solutions, and the other refers to interstitial solid solutions.

Substitutional solid solution rules

For substitutional solid solutions, the Hume-Rothery rules are as follows:

Interstitial solid solution rules

For interstitial solid solutions, the Hume-Rothery Rules are:

Solid solution rules for multicomponent systems

Fundamentally, the Hume-Rothery rules are restricted to binary systems that form either substitutional or interstitial solid solutions. However, this approach limits assessing advanced alloys which are commonly multicomponent systems. Free energy diagrams (or phase diagrams) offer in-depth knowledge of equilibrium restraints in complex systems. In essence the Hume-Rothery rules (and Pauling's rules) are based on geometrical restraints. Likewise are the advancements being done to the Hume-Rothery rules. Where they are being considered as critical contact criterion describable with Voronoi diagrams. This could ease the theoretical phase diagram generation of multicomponent systems. For alloys containing TM elements there is a difficulty in interpretation of the Hume-Rothery electron concentration rule as the e/a values for transition metals have been quite controversial for a long time and no satisfied solutions have yet emerged.